# Hybridization of first and second sound in weakly interacting Bose gases

## Sounds in superfluid 4He  Donnelly, Physics Today, 2009

• Two sound velocities in a superfluid.
• $u_2 < u_1$.
• Pressure ($u_1$) / temperature ($u_2$) wave

### Sound velocities Pitaevskii & Stringari, BEC, OUP 2003

• Two sound velocities
• $u_2 < u_1$ ?
• Hybridization at $T \approx gn$

### Nature of the sound waves ?

• Second sound is a wave where the normal and superfluid components oscillate with opposite phases.”, Hou, PRA, 2013

An interesting quantity is the Landau-Placzek ratio $\frac{c_p}{c_v} - 1$ which is always small in the case of He.

Taylor, PRA, 2009 and Landau, Fluid Mechanics

For BECs,

• Below hybridization, Landau-Placzek ratio is small.
Same behavior as in He:
Pressure ($u_1$) / Temperature ($u_2$) wave.
• Above hybridization, LP ratio is not small ⇒ very different behavior.

## Landau's two fluid model

Description of the superfluid in terms of normal and superfluid component: Donnelly, Physics Today, 2009

\begin{equation} \left\lbrace \begin{array}{l} \vec{j} = \overbrace{\rho_s \vec{v}_s}^{\text{superfluid}} + \overbrace{\rho_n \vec{v}_n}^{\text{normal}}\\ \frac{\partial \rho}{\partial t} + \mathrm{div}(\vec{j}) = 0\\ \frac{\partial \vec{j}}{\partial t} + \vec{\nabla}{P} = 0\\ \\ \\ \frac{\partial \vec{v}_s}{\partial t} + \vec{\nabla}{\mu} = 0\\ \\ \frac{\partial (\rho \tilde{s})}{\partial t} + (\rho \tilde{s})\, \mathrm{div}(\vec{v}_n) = 0\\ \end{array} \right. \end{equation}

Linear equations ⇒ look for plane wave solutions.

Plane wave solutions must satisfy a fourth order equation on sound velocity $u$:

\begin{equation} u^4 - \left( \left.\frac{\partial P}{\partial \rho}\right|_{\tilde{s}} + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v}\right) u^2 + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v} \left.\frac{\partial P}{\partial \rho}\right|_{T} = 0 \end{equation}
• Fourth order equation ⇒ 2 positive solutions.
• Valid for any superfluid system: He + BEC.
• Now we have to plug the right thermodynamic functions in this equation.
• For BECs, we will use Bogoliubov thermodynamics at low T and Hartree-Fock to lowest order at high T.

## Low temperatures

In this regime, we use Bogoliubov thermodynamics. The quasiparticles excitation spectrum is given by:

$E(\vec{p}) = \sqrt{\frac{p^2}{2m}\left( \frac{p^2}{2m} + 2 g n\right)}$

We express everything in terms of dimensionless units:

• $\tilde{p} = \frac{p}{mgn}$
• $\tilde{t} = \frac{k_B T}{gn}$

An Important parameter to have in mind is: $\eta = \frac{gn}{k_B T_c} \approx 0.03$

For example, for the free energy, we get:

\begin{equation} \frac{F(\tilde{t}\,)}{gn N}= E_0(na^3) + \eta^{3/2} \tilde{f}(\tilde{t}) \end{equation}

Express the normal density using Landau's formula:

\begin{equation} \rho_n = - \frac{1}{3} \int \frac{\mathrm{d} N_\mathbf{p}(\varepsilon)}{\mathrm{d} \varepsilon} p^2 \frac{\mathrm{d} \mathbf{p}}{(2\pi \hbar)^3} \end{equation} ## Hybridization • Hybridization at $k_BT = 0.6 gn.$
• Gap width $\propto \eta^{3/4}$.
• If $\eta = 0$, crossing.

Below hybridization, same behaviour as in liquid helium:

• First sound is an in-phase oscillation, a pressure wave.
• Second sound is an out-of-phase oscillation, a temperature wave.

## High temperatures

Ideal Bose gas thermodynamics except for $\frac{1}{\chi_T} = \frac{gn}{m}.$

For example:

$\frac{\rho_n}{\rho} = \left(\frac{T}{T_c}\right)^{3/2}$
$\frac{S}{Nk_B} = \frac{5}{2} \frac{\zeta(5/2)}{\zeta(3/2)}\left(\frac{T}{T_c}\right)^{3/2}$ • First sound branches to the adiabatic “classical” sound velocity predicted by the ideal Bose gas model above $T_c$.
• Second sound is mostly an isothermal oscillation of the superfluid component.

## [WIP] A theory for all $T < T_c$ ?

Yes ! Beliaev theory. (Giorgini, New Journal of Physics, 2010).

Self-consistent equations. For example:

$n = n_0 + \sum_{\mathbf{k}} \left[ \frac{\varepsilon_k + gn_0}{2E_k} (1+2N_E) - \frac{1}{2}\right]$

where

$\left\{\begin{array}{l} \varepsilon_k = \hbar^2 k^2 / (2m)\\ E_k = \sqrt{\varepsilon_k (\varepsilon_k + 2gn_0)} \\ N_E = \left(e^{\beta E} - 1\right)^{-1}\\ \end{array}\right.$
• Valid on a wide range of temperatures,
(except for $T \approx 0$ and $T \approx T_c$).
• Good agreement with Quantum Monte Carlo results.
• Recovers Bogoliubov at low T, and Hartree-Fock at high T.  ### But:

Some problems with the derivatives ($c_v$, $\chi_T$).

Work in progress…

## Conclusion Thanks to S. Stringari, L. P. Pitaevskii, S. Giorgini.